C₂²×#10972 #72238071

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10972

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

55037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2);(3,4);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24);(9,11)(10,14)(12,17,19,24)(13,18)(15,22,23,20)(16,21);(6,8)(9,12)(10,15)(11,17,18,24)(13,19)(14,20,21,22)(16,23)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#394064
maximal normal subgroup of	C₂²×#394064
maximal quotient of	#72238075

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2), (3,4), (5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24), (9,11)(10,14)(12,17,19,24)(13,18)(15,22,23,20)(16,21), (6,8)(9,12)(10,15)(11,17,18,24)(13,19)(14,20,21,22)(16,23));

C22×#10972 #72238071